Invariant solutions and bifurcation analysis of the nonlinear transmission line model

In this paper, the nonlinear transmission line model with the power law nonlinearity and the constant capacitance and voltage relationship is studied using Lie symmetry analysis. Corresponding to the infinitesimals obtained, using commutation relations, abelian and non-abelian Lie subalgebras are obtained. Also, using the adjoint table, a one-dimensional optimal system of subalgebra is presented. Based on this optimal system, the corresponding Lie symmetry reductions are obtained. Moreover, variety of new similarity solutions in the form of trigonometric functions, hyperbolic functions, are obtained. Corresponding to one similarity reduction, by bifurcation analysis of dynamical system, the stable and unstable regions are determined, which show the existence of soliton solutions from the nonlinear dynamics point of view. Some of the obtained solutions represented graphically and observations are also discussed.


Introduction
The concept of nonlinearity is observed in many areas of physics [1][2][3][4][5][6][7][8][9][10][11][12]. To understand the nonlinear phenomena, which are described by nonlinear partial differential equations (NLPDEs), we try to obtain their exact analytical solutions if it exists; otherwise, we try to obtain the numerical solutions. Recently, a lot of work has been done to study the exact solutions of these NLPDEs. Some of them include the Lie symmetry method [13,14], Direct method for symmetries [15], non-classical symmetry method [16], Bäcklund transformation method [17,18], solitary wave ansatz method [19,20], Hirota's bilinear method [21,22], the modified simple equation method [23], the G G expansion method [24,25] and so on. Some authors also developed some numerical techniques to obtain the approximate solutions of NLPDEs [26][27][28] Among all the methods in the literature, the Lie symmetry method [13,14,29] is one of the most effective and powerful methods for finding the travelling and nontravelling wave solutions of NLPDEs. Using Lie symmetries, it is possible to reduce the number of independent variables and further, to reduce the order of ordinary differential equations, thus making them easier to solve. Once the symmetry group is constructed, it can be used to determine new solutions of the NLPDEs from the known ones. Only limitation of this method is that calculations involved are cumbersome. For higher-order NLPDEs, calculations become more tedious. But nowadays, software like Mathematica, Maple, REDUCE, Maxima and wxMaxima makes it easy to apply this method. Some of the recent work in this field can be seen in [30][31][32][33].
In electronics and communication engineering, a transmission line is a cable designed to conduct the currents with a frequency high enough that their wave nature must be taken into account. In this paper, we consider a nonlinear transmission line (NLTL) [34] of the following form: where C and L denote the capacitance (depends on voltage v) and inductance, respectively. Afshari et al. [35] proposed and obtained travelling wave and numerical solutions of one-and two-dimensional non-uniform NLTL equations. El-Borai et al. [34] used extended tanh equation method to obtain the soliton solutions of (1.1) for the relation C(v) = C 0 (1 − bv). Mostafa [6] obtained analytical solutions of NLTL equation using improved tanh and sech methods for capacitance voltage (C-V) relationships C(v) = C 0 (1 − bv) and In this work, Lie point symmetry analysis [36,37] of a class of nonlinear transmission line model, with respect to capacitance and voltage relationship, is discussed. Let us approximate the capacitor's voltage dependence in the form of power law nonlinearity, linear and constant relations as follows: and respectively. As mentioned above, some special cases of NLTL model without power law nonlinearity were already discussed by several authors. The main purpose of this paper is to obtain symmetry reductions and exact solutions to the nonlinear transmission lines for the capacitor's voltage dependence in the forms (1.2)-(1.3). To the best of our knowledge, in this work, NLTL for power law nonlinearity is first time considered for Lie symmetry analysis and for obtaining the analytical exact solutions. So far, some authors [6,7,34,35] have considered the particular case of NLTL (1.4) for n = 0, 1 or n = 2 and obtained their travelling wave solutions. We have considered the general form of NLTL with power law nonlinearity and obtained travelling wave as well as non-travelling wave solutions. In the literature, even bifurcation analysis of NLTL with power law nonlinearity has not been performed by any author. So, this shows the novelty and originality of this work.
The paper is organized as follows. Firstly, symmetries of nonlinear transmission lines (1.4)-(1.5) are obtained using the Lie classical method. Then, corresponding to the optimal system, the symmetry reductions and exact solutions of the equations are obtained. Some graphical representations of obtained solutions are also given. Furthermore, phase plane analysis, corresponding to one of similarity reduction, is also performed.

Lie symmetry analysis
Let us first consider the Lie group of point transformations where ξ, τ , and η are infinitesimals and is very small parameter; thus, 1. Let be the infinitesimal generator of the Lie group of point transformations (2.6). Let Definition 2.1 (Lie Algebra) [38] A Lie algebra, L, is a vector space over some field with one more operation, commutation [, ], connecting elements of L such that the following properties are satisfied: where η tt , η x x , η x x x x are prolonged infinitesimals acting on an extended space that contains all the derivatives of the dependent variable i.e., v tt , v x x , and v x x x x . The infinitesimal generator of a point symmetry admitted by Eq. (1.4) is of the form: The values of infinitesimals up to the fourth prolongation [38] are given by Therefore, computing and substituting the values of η tt ,η x x and η x x x x in Eq. (2.9), we get the symmetry determining equations. Solving the determining equations, the following expressions for the infinitesimals ξ, τ, η are obtained: where c 1 , c 2 , c 3 are arbitrary parameters. The point symmetry generators admitted by Eq. (1.4) are given by The commutator table of the generators is given in Table 1.
From the table, it can be seen that the generators form a Lie algebra.
From commutator table, one can see that X 1 , X 2 generate abelian Lie subalgebra and X 1 , X 3 forms nonabelian Lie subalgebra [39]. The adjoint representation [14] is given by: The three-dimensional Lie algebra L 3 is solvable, and the adjoint table of the generators is given in Table 2. [14] of Lie algebras is provided by:

Theorem 2.3 An optimal system
Proof Consider the symmetry algebra of Eq. (1.4) whose adjoint representation is determined in Table 2.
Let us consider the nonzero vector in the form (2.14) We need to simplify this vector field by using suitable adjoint maps. First, assume that a 3 = 0 and assume a 3 = 1. Now if we apply on X by Ad(exp(a 1 X 3 ))X , using Table 2, the coefficient of X 1 will vanish as follows: Ad(e a 1 X 3 )X = a 2 X 2 + X 3 .
(2.15) It cannot be reduced further by applying adjoint operations. So, X is equivalent to a 2 X 2 + X 3 under the adjoint representation. The remaining subalgebras can be spanned by vectors of the form X with a 3 = 0. If a 2 = 0, we may take a 2 = 1, and then, we have (2.16) Table 1 Commutator table We can further act on X by the group generated by X 3 , which has the effect of scaling the coefficient of X 1 : Thus, any sub-algebra spanned by X with a 3 = 0, a 2 = 0 is equivalent to one spanned by either a 1 e X 1 + X 2 , where a 1 is arbitrary and may take values +1, −1 or 0. Further, by assuming a 3 = a 2 = 0 and a 1 = 1, we are left with subalgebra X 1 .

Determination of Lie symmetries for equation (1.5)
As procedure mentioned above, for Eq. (1.5), we get rich class of symmetries and the infinitesimals ξ, τ and η are obtained as follows: where C 1 , C 2 , . . . , C 11 are arbitrary constants. The eleven-dimensional symmetry algebra admitted by Eq. (1.5) is given by (2. 19) In this case, the nonzero commutator relations are obtained as follows: So, Lie algebra generated by vector fields X i , i = 1, 2, . . . , 11, is non-abelian. For reduction, we will consider the following linear combinations of vector fields: where μ, λ, C 3 , C 5 , C 6 , C 9 are arbitrary constants.

Symmetry reductions
In this section, the symmetry reductions in nonlinear transmission line Eqs. (1.4)-(1.5) are obtained.

Similarity reductions for equation (1.4)
As explained in Sect. 2.1, optimal system of vector fields for Eq. (1.4) is: For vector field X 3 + λX 2 Corresponding to this vector field, solving the characteristic equation, we have the following similarity variables where s is new independent variable and F is new dependent variable. Using (3.24) in Eq. (1.4), we obtain the following ordinary differential equation (ODE) where denotes the derivative with respect to s.
For vector field X 2 + μX 1 For vector field X 2 + μX 1 , the similarity variables are where ξ and G are corresponding new variables. Using (3.26) in (1.4), we have (3.27) where denotes the derivative with respect to ξ .

For vector field X 1
Corresponding to this vector field, the similarity variables are as follows: where denotes derivative with respect to τ .

Similarity reduction for equation (1.5)
For vector field X 1 + μX 2 + λX 7 Solving the characteristic equations for the vector field (2.21), we have the following similarity variables where ξ and F are new independent variable and dependent variable, respectively. Using (3.30) in (1.5), we have the following ODE For vector field X 1 + μX 2 + C 3 X 3 + C 9 X 9 For vector field (2.22), we have the following similarity variables of Eq.

Exact solutions
In this section, we will obtain the exact solutions of Eqs. (1.4)-(1.5).

For equation (1.4)
For finding the solutions of (1.4), firstly we will solve Eq. (3.25). Let us assume the solution of Eq. (3.25) in the following form where k and p have to be determined. Using (4.36) in (3.25) and simplifying, we get the following values For solving Eq. (3.27), substitute we obtain the following ODE Now integrating (4.40) twice and taking constant of integration equal to zero, we have Balancing the highest derivative term and nonlinear term, we have m Clearly m is integer only if n = 1 or n = 2. Case n = 1 will be discussed separately. For all other values of n, let us apply the transformation Using (4.43), Eq. (4.41) reduces to Now for obtaining solutions corresponding to ODE (4.44), we will apply extended ( G G )-expansion method [24].

Method description
Let us consider a general nonlinear PDE with constant coefficients as follows: where u = u(x, t) is an unknown function; F is polynomial in u(x, t) and its partial derivatives, in which the highest derivative term and nonlinear terms are involved. The following are main steps involved.
Step 1. To reduce Eq. (4.45) into ordinary differential equation (ODE), we consider a variable ξ such that where μ is speed of travelling wave. Using (4.46) into (4.45), we have where denotes derivative with respect to ξ .
Step 3. Let us consider the solution of Eq. (4.47) in the following form: 1, . . . , m) are arbitrary constants to be determined later, ρ = ±1, m is positive integer, and G = G(ξ ) satisfies the following ODE where μ 1 is to be determined.
Step 4. Now determine the value of m by homogeneous balance between the highest derivative term and highest nonlinear term for Eq. (4.47).
Step 5. Substitute where C 1 and C 2 are two arbitrary constants. Family 2. When μ 1 > 0, then we have trigonometric type solutions as follows: where C 1 and C 2 are two arbitrary constants. Family 3. If μ 1 = 0, then we have rational type solutions as follows: where C 1 and C 2 are two arbitrary constants. Similar type of solutions for the nonlinear Schrödinger equation in the form Kuznetsov breather, the Akhmediev breather, and the Peregrine solution has also been obtained by some of the authors . Now we will apply the method to ODE (4.44). Balancing the highest order derivative term and nonlinear term in Eq. (4.44), we have m = 2.
So, as per algorithm for the extended ( G G ) expansion method, value of solution W (σ ) of Eq. (4.44) is given by where G satisfy Eq.

Particular cases
For particular values of C 1 and C 2 in aforementioned solutions, soliton, periodic, and complex solutions can be obtained when parameters take special values as follows:

Observation
One can observe that: (i) As ξ → ±∞, graph of solution is parabolic as shown in Fig. 2(a). (ii) For transient value of ξ , solution is represented by bell shaped solitons as shown in Fig. 2(b).
Now, let us obtain the solution of ODE (3.35). Using Maple, we obtain the following solution of the ODE (3.35) Corresponding solution of (1.5) is given by (4.79) Solution (4.78) is represented in Fig. 3.

Observation
One can observe that: (i) With taking coefficient of ξ nonzero, the graph of the solution is parabolic as shown in Fig. 3(a). (ii) For taking coefficient of ξ equal to zero, the graph of the solution is periodic as shown in Fig. 3(b). (iii) Also, with the increasing the value of μ, the amplitude of travelling wave remains same, but width increases as shown in Fig. 3(b).

Bifurcation analysis
In this section, the transmission line model with power law nonlinearity will be studied using the dynamical system approach. The phase portraits will be displayed.
1, C 2 = 0, A 2 = 0, L = 1.2 and μ = 0.5, μ = 0.8 are shown corresponding to red, blue colors, respectively 5.1 Bifurcation phase portraits and qualitative analysis Introducing the notation X = V, Y = V , let us reduce Eq. (4.41) to the autonomous system: where a 0 = δ 2 12 and a 1 = μ 2 C 0 L n+1 . The system (5.80) is a Hamiltonian system with the following Hamiltonian function: If the system is not Hamiltonian, then there is no closed orbit in the phase portrait. So, in that case, the system will not have solitary and periodic wave solutions. For obtaining the phase portrait of (5.80), set When n is even number and a 1 > 0, f (X ) has three zeros, X − , X 0 and X + given as follows: When n is odd number, f (X ) has two zeros, X 0 and X * given as follows: Let (X i , 0) be the critical point of (5.80), then the corresponding characteristic values of linearized system (5.80) at the singular points (X i , 0) are given by Using the qualitative theory of dynamical system, we know that So, the bifurcation phase portraits of the system     The qualitative theory of dynamical systems suggests that a smooth homoclinic orbit of a travelling wave equation results in a smooth solitary wave solution of a partial differential equation. A smooth heteroclinic orbit of a travelling wave equation results in a smooth kink wave solution or an unbounded wave solution. Similarly, a periodic orbit of a traveling wave equation results in a periodic travelling wave solution of a partial differential equation. So, we have the following propositions.

Travelling wave solutions
Firstly, let us consider the case when n is even and na 0 > 0, we obtain the explicit expressions for travelling wave solutions for (1.4). From phase portrait, we have two special orbits P1 and P1 * which have the same value of Hamiltonian as that of center point (0, 0). In (X, Y ) plane, the expressions of the orbits are given by where a 0 = δ 2 12 and a 1 = μ 2 C 0 L n+1 .

Conclusion
In this work, nonlinear transmission line model (NLTL) with arbitrary power law nonlinearity and the constant capacitor voltage dependence has been studied through application of Lie symmetry analysis. For both the cases, i.e., C(v) = C 0 (1 − bv) and C(v) = C 0 of NLTL, infinitesimal symmetries are obtained and partial differential equations are reduced to ordinary differential equations (ODEs). Furthermore, variety of new exact solutions in the form of trigonometric functions, hyperbolic functions solutions, are also obtained.
Additionally, corresponding to one reduction in NLTL with power law nonlinearity, phase plane analysis is carried out. This results in various propositions where various solution existence structures are examined that depend upon the parameter values. Finally, some more solutions of NLTL model in the form of periodic singular waves and singular soliton waves are obtained. Graphical representation of some of solutions is also given and observed effect of parameters in graphs is discussed.
Data Availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of interest
The authors declare that they have no conflict of interest.